After all possible Bitcoins are created, won't transaction fees from miners validating transactions eat away from the total and eventually consume all Bitcoins?

Right now, because new Bitcoins are still being created, kind of like planned inflation, mining fees are adding Bitcoins to the circulation.

But after all Bitcoins are created, it seems that an infinite number of transactions would transfer all Bitcoins via fees to miners, self-defeating the currency.

I wonder if this problem, if real, is another reason why a little bit of inflation is a good thing, since some types of fiat currency transactions consume fees (e.g., credit card transactions) but others do not (cash, direct bank transfers, ATM withdrawals from member bank systems) or don't per transaction ( flat monthly checking account fees). It seems a little bit of inflation prevents fees from taking an ever greater percentage of a currency.

I'm skating on very thin ice with my limited understanding of Bitcoin and currency in general, and that's why I'm posting here.

Fees are a mammoth devil in the details of the world economy, maybe including Bitcoin.

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MAJOR EDIT ADDITION

There seems to be a misunderstanding about my question. I'm not questioning the value of miners. I'm wondering about the mathematics of compensating miners after all 21 million bitcoins are created. This isn't a moral judgement. Miners are creating bitcoins now, providing a valuable service. But since there's a finite number of bitcoins, 21 million (?), after they're all mined won't miners still be necessary to process all transactions? But instead of creating new bitcoins, after the maximum of 21 million coins are created, won't miners then be taking bitcoins from other people instead of creating new ones? Just mathematically, after all 21 million bitcoins are created, wouldn't an infinite number of transactions transfer all bitcoins to miners? I'm just wondering about the math theory of mining after all 21 million bitcoins are created.

4 Answers
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Miners have expenses. You need to pay your power bill. You need to pay your employees. You need to pay for new ASICs when the old ones inevitably become obsolete. You need to pay off your initial investment in ASICs. All of these things cost money.

You might object that miners wouldn't be doing this unless revenue exceeded expenses. This is true, but it's also true of every other profession. Therefore, we should view 'miners will eventually have all the money' the same way we view 'dentists will eventually have all the money.'

In general, people don't acquire money for the sake of having money. It's usually in service of some other kind of goal, like financial security, providing for someone else, or buying food. Sitting on a hoard of money that you cannot spend accomplishes none of these goals.

Technically yes, but miners themselves can then sell those Bitcoins back into the pool for non-miners to pay for their own goods and services. While this brings up an interesting point about an eventual "miner elite", this is far in the future and for now a non-issue. Ultimately, some other coin may need to become dominant to stop this centralization you describe.

As others have pointed out, Scrypt coins democratize mining, and proof of stake type coins also offer a way out of that sort of dilemma!